{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Notation\n",
    "\n",
    ":label:`chap_notation`\n",
    "\n",
    "\n",
    "The notation used throughout this book is summarized below.\n",
    "\n",
    "\n",
    "## Numbers\n",
    "\n",
    "* $x$: A scalar\n",
    "* $\\mathbf{x}$: A vector\n",
    "* $\\mathbf{X}$: A matrix\n",
    "* $\\mathsf{X}$: A tensor\n",
    "* $\\mathbf{I}$: An identity matrix\n",
    "* $x_i$, $[\\mathbf{x}]_i$: The $i^\\mathrm{th}$ element of vector $\\mathbf{x}$ \n",
    "* $x_{ij}$, $[\\mathbf{X}]_{ij}$: The element of matrix $\\mathbf{X}$ at row $i$ and column $j$\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "## Set Theory\n",
    "\n",
    "\n",
    "* $\\mathcal{X}$: A set\n",
    "* $\\mathbb{Z}$: The set of integers\n",
    "* $\\mathbb{R}$: The set of real numbers\n",
    "* $\\mathbb{R}^n$: The set of $n$-dimensional vectors of real numbers\n",
    "* $\\mathbb{R}^{a\\times b}$: The set of matrices of real numbers with $a$ rows and $b$ columns\n",
    "* $\\mathcal{A}\\cup\\mathcal{B}$: Union of sets $\\mathcal{A}$ and $\\mathcal{B}$\n",
    "* $\\mathcal{A}\\cap\\mathcal{B}$: Intersection of sets $\\mathcal{A}$ and $\\mathcal{B}$\n",
    "* $\\mathcal{A}\\setminus\\mathcal{B}$: Subtraction of set $\\mathcal{B}$ from set $\\mathcal{A}$\n",
    "\n",
    "\n",
    "## Functions and Operators\n",
    "\n",
    "\n",
    "* $f(\\cdot)$: A function\n",
    "* $\\log(\\cdot)$: The natural logarithm\n",
    "* $\\exp(\\cdot)$: The exponential function        \n",
    "* $\\mathbf{1}_\\mathcal{X}$: The indicator function\n",
    "* $\\mathbf{(\\cdot)}^\\top$: Transpose of a vector or a matrix             \n",
    "* $\\mathbf{X}^{-1}$: Inverse of matrix $\\mathbf{X}$\n",
    "* $\\odot$: Hadamard (elementwise) product \n",
    "* $[\\cdot, \\cdot]$: Concatenation\n",
    "* $\\lvert \\mathcal{X} \\rvert$: Cardinality of set $\\mathcal{X}$\n",
    "* $\\|\\cdot\\|_p$: $\\ell_p$ norm                                 \n",
    "* $\\|\\cdot\\|$: $\\ell_2$ norm        \n",
    "* $\\langle \\mathbf{x}, \\mathbf{y} \\rangle$: Dot product of vectors $\\mathbf{x}$ and $\\mathbf{y}$  \n",
    "* $\\sum$: Series addition                        \n",
    "* $\\prod$: Series multiplication                  \n",
    "\n",
    "\n",
    "## Calculus\n",
    "\n",
    "* $\\frac{dy}{dx}$: Derivative of $y$ with respect to $x$        \n",
    "* $\\frac{\\partial y}{\\partial x}$: Partial derivative of $y$ with respect to $x$\n",
    "* $\\nabla_{\\mathbf{x}} y$: Gradient of $y$ with respect to $\\mathbf{x}$  \n",
    "* $\\int_a^b f(x) \\;dx$: Definite integral of $f$ from $a$ to $b$ with respect to $x$\n",
    "* $\\int f(x) \\;dx$: Indefinite integral of $f$ with respect to $x$\n",
    "\n",
    "## Probability and Information Theory\n",
    "\n",
    "* $P(\\cdot)$: Probability distribution                                \n",
    "* $z \\sim P$: Random variable $z$ has probability distribution $P$ \n",
    "* $P(X \\mid Y)$: Conditional probability of $X \\mid Y$        \n",
    "* $p(x)$: Probability density function\n",
    "* ${E}_{x} [f(x)]$: Expectation of $f$ with respect to $x$  \n",
    "* $X \\perp Y$: Random variables $X$ and $Y$ are independent\n",
    "* $X \\perp Y \\mid Z$: Random variables  $X$  and  $Y$  are conditionally independent given random variable $Z$\n",
    "* $\\mathrm{Var}(X)$: Variance of random variable $X$\n",
    "* $\\sigma_X$: Standard deviation of random variable $X$\n",
    "* $\\mathrm{Cov}(X, Y)$: Covariance of random variables $X$ and $Y$\n",
    "* $\\rho(X, Y)$: Correlation of random variables $X$ and $Y$\n",
    "* $H(X)$: Entropy of random variable $X$\n",
    "* $D_{\\mathrm{KL}}(P\\|Q)$: KL-divergence of distributions $P$ and $Q$\n",
    "\n",
    "\n",
    "\n",
    "## Complexity\n",
    "\n",
    "* $\\mathcal{O}$: Big O notation\n",
    "\n"
   ]
  }
 ],
 "metadata": {
  "language_info": {
   "name": "python"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}